Final answer:
The total income after working for various numbers of days in a job where the pay doubles each day is calculated by the geometric series formula. For 3 days, the sum is $0.07; for 5 days, it's $0.31; for 10 days, it's $10.23; and for 20 days, a surprisingly large total of $10,485.75 is reached due to exponential growth. A) 3 days
Step-by-step explanation:
The scenario presented is a classic example of a geometric progression where the daily wage doubles each day, which is a common mathematical concept covered in middle school math curriculum. The wage progression follows the pattern of 2n-1 where n is the day number. The total income for a certain number of days is the sum of a geometric series.
- For 3 days: $0.01 + $0.02 + $0.04 = $0.07
- For 5 days: $0.01 + $0.02 + $0.04 + $0.08 + $0.16 = $0.31
- For 10 days: The total is $0.01(210 - 1) / (2 - 1) = $10.23
- For 20 days: The total is $0.01(220 - 1)/(2 - 1) = $10,485.75
To calculate the total income precisely, we apply the formula for the sum of the first n terms of a geometric series: Sn = a(1 - rn) / (1 - r), with 'a' being the first term and 'r' the common ratio. Here, 'a' is $0.01, and 'r' is 2. It's important to note that for practical and real-life applications, understanding the concept of exponential growth, similar to how interest accumulates in a bank account or the growth rate when given a pay raise, is essential.